Are all those apps on your itunes library on your computer or just on your old iphone?
Plug the iphone 3GS, click the apps tab and select the sync apps checkbox.
Then on the bottom right hit Apply then sync.
It will take a while to sync them all back in so give it 5-10 minutes depending how many and how big your apps are.

When I backed up my 3g I assumed the apps backed up as well? Because when I plug in my new 3gs and click on the apps tab, there are no apps there for me to sync. What am I missing here? I really appreciate your help!!!

When I backed up my 3g I assumed the apps backed up as well? Because when I plug in my new 3gs and click on the apps tab, there are no apps there for me to sync. What am I missing here? I really appreciate your help!!!

No, Apps never backup. You gotta sync them back from your computers itunes library.
You needed to transfer you apps into your itunes library.
So when you open up itunes and click on the Apps icon under Library, music,movies, podcasts etc.. there's nothing there?
No apps?
You need to plug your iphone 3G back if you have it available and right click under devices on your iphone 3G icon and select "Transfer Purchases"
That will copy/transfer all your apps from your iphone 3G to your itunes library and then you will be able to sync them back to the 3GS.
Or you can redownload them all again for free with the same itunes account login.

I've seen the reply from the demi-god and tried to follow the information but this doesn't work for me as the apps can be seen in the library but will not transfer with the information over. If an app is free of charge I don't think it will count as a 'transfer purchase'. Does this mean I have to say goodbye to my apps? I've spent a whole hour on the phone with apple and they cannot help! By the way, where is the app check box? I cannot find it!
Thanks Susan